(a) The curve is convex for . With and , show that and that recovers the original…

PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · Thermodynamic potentials and the formal structure of thermodynamics · Thermodynamic potentials and the formal structure of thermodynamics

Thermal Physics, Ch Thermodynamic potentials, Legendre transforms and Maxwell relations

Problem

(a) The curve

y = x^3

is convex for

x > 0

. With

s = \mathrm{d}y/\mathrm{d}x

and

\phi(s) = y - sx

, show that

\phi(s) = -2(s/3)^{3/2}

and that

x = -\mathrm{d}\phi/\mathrm{d}s

recovers the original variable. (b) Starting from

\mathrm{d}H = T\,\mathrm{d}S + V\,\mathrm{d}p

and the equality of mixed partials, derive the Maxwell relation connecting

(\partial T/\partial p)_S

and

(\partial V/\partial S)_p

. Then state, in one sentence, the rule that tells you which potential to differentiate when the two variables held fixed are given. (c) A change takes place in an open beaker whose contents are held at the fixed temperature

T

of the room and the fixed pressure

p

of the atmosphere. The contents receive heat

Q

, the room's entropy changes by

-Q/T

, and the entropy of contents plus room cannot decrease. Show that the change requires

\Delta G \le 0

for the contents, where

G = U + pV - TS

. Say in one sentence why a gas sealed in a rigid box at fixed temperature is governed by

F

and not by

G

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